According to the quantum Cramér-Rao bound, a higher accuracy is obtained for small variances, which correspond to the largest values of the quantum Fisher information. Thus, the primary aim of all quantum metrology protocols is to reach the smallest value of the variance. However, recent studies have revealed broad connections between quantum Fisher information and other aspects of quantum mechanics, including characterization of quantum correlations [1]}, quantum phase transition [2]}, entanglement control [3]}, quantum speed limit [4]} and quantum thermodynamics [5]}. These links indicate that it is more than a concept in quantum metrology, but rather a fundamental quantity in quantum mechanics. Besides, it is well known that the achievable phase estimation accuracy, in the classical optical interferometry theory, is bounded by the shot noise. Then the phase uncertainty can only reach the accuracy scale \(1/\sqrt{N}\) , where \(N\)  stands for the average photon number of the light field. This phase uncertainty limit is called the standard quantum limit [6]}. Utilizing the non-classical states of light, which in principle contains a non-zero amount of correlations, the quantum metrology theory allows to beat the shot noise and the accuracy of the phase shift estimation in an optical interferometer. The use of NOON states for example, which are maximally entangled states, can lead to a scaling of the precision \(1/N\) . This bound being called the Heisenberg limit [7]}. Several reasons explain why quantum systems are able to outperform classical devices, and perhaps most importantly, the classical measurement boundary can be exceeded using quantum systems.
